Respiratory rate (RR) is a physiological signal that is vital for many health and clinical applications. This paper presents RespWatch, a wearable sensing system for robust RR monitoring on smartwatches with Photoplethysmography (PPG). We designed two novel RR estimators based on signal processing and deep learning. The signal processing estimator achieved high accuracy and efficiency in the presence of moderate noise. In comparison, the deep learning estimator, based on a convolutional neural network (CNN), was more robust against noise artifacts at a higher processing cost. To exploit their complementary strengths, we further developed a hybrid estimator that dynamically switches between the signal processing and deep learning estimators based on a new Estimation Quality Index (EQI). We evaluated and compared these approaches on a dataset collected from 30 participants. The hybrid estimator achieved the lowest overall mean absolute error, balancing robustness and efficiency. Furthermore, we implemented RespWatch on commercial Wear OS smartwatches. Empirical evaluation demonstrated the feasibility and efficiency of RespWatch for RR monitoring on smartwatch platforms.